[was@modular was]$ [was@modular was]$ Magma V2.8-2      Wed Dec  5 2001 16:45:03 on modular  [Seed = 571509591]
Type ? for help.  Type <Ctrl>-D to quit.

Loading startup file "/home/was/magma/local/emacs.m"

Loading "/home/was/magma/local/init.m"
> M := ModularSymbols(DirichletGroup(37).1,2);
> M := ModularForms(DirichletGroup(37).1,2);
> M;
Space of modular forms on Gamma_1(37) with character all conjugates of [$.1], weight 2, and dimension 4 over Integer Ring.
> M := ModularForms(DirichletGroup(100).1,4);
> M;
Space of modular forms on Gamma_1(100) with character all conjugates of [$.1], weight 4, and dimension 0 over Integer Ring.
> M := ModularForms(DirichletGroup(389).1,4);
> M;
Space of modular forms on Gamma_1(389) with character all conjugates of [$.1], weight 4, and dimension 98 over Integer Ring.
> M := ModularForms(DirichletGroup(389).1,30);
> M;
Space of modular forms on Gamma_1(389) with character all conjugates of [$.1], weight 30, and dimension 944 over Integer Ring.
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> eps := DirichletGroup(27,GF(7)).1; M := ModularForms( eps, 3);

>> eps := DirichletGroup(27,GF(7)).1; M := ModularForms( eps, 3);
                                                       ^
Runtime error in 'ModularForms': The base ring of argument 1 must be the rationals or cyclotomic.
> eps := DirichletGroup(27).1; M := ModularForms(eps, 3);
> M;
Space of modular forms on Gamma_1(27) with character all conjugates of [$.1], weight 3, and dimension 9 over Integer Ring.
> M7 := BaseExtend(M,GF(7));
> M;
Space of modular forms on Gamma_1(27) with character all conjugates of [$.1], weight 3, and dimension 9 over Integer Ring.
> Basis(M);
[
1 + O(q^8),
q + O(q^8),
q^2 + O(q^8),
q^3 + O(q^8),
q^4 + O(q^8),
q^5 + O(q^8),
q^6 + O(q^8),
q^7 + O(q^8),
O(q^8)
]
> Basis(M7);
[
1 + O(q^8),
q + O(q^8),
q^2 + O(q^8),
q^3 + O(q^8),
q^4 + O(q^8),
q^5 + O(q^8),
q^6 + O(q^8),
q^7 + O(q^8),
O(q^8)
]
> S7 := CuspidalSubspace(M7);
> S7;
Space of modular forms on Gamma_1(27) with character all conjugates of [$.1], weight 3, and dimension 3 over Finite field of size 7.
> N7 := Newforms(S7);

>> N7 := Newforms(S7);
                 ^
Runtime error in 'Newforms': Argument 1 must have characteristic 0.
> N7 := Newforms(CuspidalSubspace(M));
> N7;
[* [*
q + 4*q^4 - 13*q^7 + O(q^8)
*], [*
q + a*q^2 - 5*q^4 - a*q^5 + 5*q^7 + O(q^8),
q + b*q^2 - 5*q^4 - b*q^5 + 5*q^7 + O(q^8)
*] *]
> Reductions;
Intrinsic 'Reductions'

Signatures:

(<ModFrmElt> f, <RngIntElt> p) -> List

The mod p reductions of the modular form f. Because of denominators, the list of reductions can be empty.


> Newform(M,1);
q + 4*q^4 - 13*q^7 + O(q^8)
> Newform(M,2);
q + a*q^2 - 5*q^4 - a*q^5 + 5*q^7 + O(q^8)
> Reductions(Newform(M,1), 7);
[* [*
q + 4*q^4 + q^7 + O(q^8)
*] *]
> Reductions(Newform(M,2), 7);
[* [*
q + $.1^44*q^2 + 2*q^4 + $.1^20*q^5 + 5*q^7 + O(q^8),
q + $.1^20*q^2 + 2*q^4 + $.1^44*q^5 + 5*q^7 + O(q^8)
*] *]
> R := Reductions(Newform(M,2), 7);
> f := R[1][1];
> f;
q + $.1^44*q^2 + 2*q^4 + $.1^20*q^5 + 5*q^7 + O(q^8)
> Parent(f);
Space of modular forms on Gamma_1(27) with character all conjugates of [$.1], weight 3, and dimension 2 over Finite field of size 7^2.
> BaseRing(Parent(f));
Finite field of size 7^2
> CoefficientRing(f);
Finite field of size 7^2
> IsNewform(f);
false
> IsNewform(Newform(M,2));
true
> IsNewform;
Intrinsic 'IsNewform'

Signatures:

(<ModFrmElt> f) -> BoolElt

True if f was created using Newform.


> FieldOfFractions(GF(7));
Finite field of size 7
> ;
> g := Newform(M,2);
> CoefficientField(g);
Number Field with defining polynomial x^2 + 9 over the Rational Field
> Order;
Intrinsic 'Order'

Signatures:

(<RngIntElt> a, <RngIntElt> m) -> RngIntElt

The least integer k with 0 < k < m such that a^k = 1 mod m, (0 if a & m not coprime) [m > 1]

(<FldFinElt> x) -> RngIntElt
(<RngIntResElt> x) -> RngIntElt
(<GrpPCElt> x) -> RngIntElt
(<GrpPermElt> x) -> RngIntElt
(<GrpGPCElt> x) -> RngIntElt
(<GrpAutoElt> x) -> RngIntElt

The order of the group element x

(<GrpAbElt> x) -> RngIntElt

The order of the group element x, or zero if it has infinite order

(<MtrxS> X) -> RngIntElt, BoolElt
[
Proof: BoolElt
]

The order of the invertible matrix X; if Proof is false, then difficult integer factorizations are not attempted and the first return value O may be only a multiple of the order of X; in any case the second return value indicates whether O is known to be the exact order of X

(<AlgChtrElt> M) -> RngIntElt

The cardinality of M

(<GrpFP> G) -> RngIntElt
[
Strategy: Any, 
Compact: RngIntElt, 
CosetLimit: RngIntElt, 
FillFactor: RngIntElt, 
Style: "C" | "CR" | "Cr" | "R" | "R_CR" | "Rc" | "Rt", 
CTFactor: RngIntElt, 
RTFactor: RngIntElt, 
RelationsInSubgroup: RngIntElt, 
Print: Any, 
Messages: RngIntElt, 
TimeLimit: RngIntElt, 
Workspace: RngIntElt, 
Mendelsohn: BoolElt, 
RowFilling: BoolElt, 
PathCompression: BoolElt, 
Lookahead: RngIntElt, 
PrefDefMode: RngIntElt, 
PrefDefSize: RngIntElt, 
DeductionMode: RngIntElt, 
DeductionSize: RngIntElt, 
LoopLimit: RngIntElt, 
Grain: RngIntElt, 
Hard: BoolElt, 
SubgroupRelations: RngIntElt, 
Time: RngIntElt, 
UseRewrite: BoolElt, 
MinIndex: RngIntElt, 
MaxIndex: RngIntElt
]

The order of the group G, or zero if it cannot be determined

(<GrpPC> G) -> RngIntElt
(<GrpAut> G) -> RngIntElt
(<GrpCayg> G) -> RngIntElt
(<GrpMat> G) -> RngIntElt
(<GrpPerm> G) -> RngIntElt
(<GrpAuto> G) -> RngIntElt
(<GrpGPC> x) -> RngIntElt

The order of the group G

(<MonRWS> M) -> RngIntElt

The order of M

(<GrpAb> G) -> RngIntElt

The order of the group G, or zero if it is infinite

(<GrpPCpQuotientProc> P) -> RngIntElt

The order of the group currently defined by P

(<Grph> G) -> RngIntElt

The number of vertices p of the (p, q) graph G

(<RngOrd> O, <AlgMatElt> T, <RngIntElt> d) -> RngOrd
[
Check: BoolElt
]

The equation order with basis Basis(O)*T/d

(<SubGrpLatElt> C) -> RngIntElt

The order of the elements of the conjugacy class of subgroups C

(<SchGrpEll> E) -> RngIntElt
(<SetPtEll> E) -> RngIntElt

The order of the group of rational points on the elliptic curve E defined over a finite field

(<PtEll> P) -> RngIntElt

The order of the elliptic curve point P (0 if order is infinite)

(<Dsgn> D) -> RngIntElt

The order of the design D

(<Plane> P) -> RngIntElt

The order of the projective plane P

(<RngOrdIdl> I) -> RngOrd
(<RngOrdFracIdl> I) -> RngOrd
(<RngFunOrdIdl> I) -> RngFunOrd

The order that the ideal I is over

(<Aut> a) -> RngIntElt

The order of the automorphism a

(<QuadBinElt> f) -> RngIntElt

The order of the quadratic form f

(<SeqEnum[FldAlgElt]> l) -> RngOrd
[
Verify: BoolElt, 
Order: BoolElt
]

The minimal order containing all elements of l

(<GrpAbGen> GA) -> RngIntElt

The order of the generic abelian group GA

(<GrpAbGenElt> g) -> RngIntElt
[
ComputeGroupOrder: BoolElt, 
BSGSLowerBound: RngIntElt, 
BSGSStepWidth: RngIntElt
]

The order of the group element g which is an element of the GrpAbGen GA. If ComputeGroupOrder is true, the group order is computed first, and the element order is computed using this knowledge. If ComputeGroupOrder is false, the element order is computed using an improved baby-step giant-step algorithm. ComputeGroupOrder is true by default. BSGSLowerBound (a lowerbound on the element order) and BSGSStepWidth (the step-width) are two parameters which may be passed to the baby-step giant-step algorithm. 

(<GrpAbGenElt> g, <RngIntElt> l, <RngIntElt> u) -> RngIntElt
[
Alg: "PollardRho" | "Shanks", 
UseInversion: BoolElt
]

Computes the order of g using either the generic Shanks or a Pollard-Rho variant. The search space must be bounded by l and u. Setting UseInversion halves the search space. By default, Alg is Shanks and UseInversion is false. 

(<GrpAbGenElt> g, <RngIntElt> l, <RngIntElt> u, <RngIntElt> n0, <RngIntElt> m) -> RngIntElt
[
Alg: "PollardRho" | "Shanks", 
UseInversion: BoolElt
]

Computes the order of g using either the generic Shanks or a Pollard-Rho variant. The search space must be bounded by l and u and n0 and m are such that |g| (or the group order) is n0 mod m. Setting UseInversion halves the search space. By default, Alg is Shanks and UseInversion is false. 

(<JacHypPt> P, <RngIntElt> l, <RngIntElt> u) -> RngIntElt
[
Alg: "PollardRho" | "Shanks", 
UseInversion: BoolElt
]

Computes the order of P using either the generic Shanks or a Pollard-Rho variant. The search space must be bounded by l and u. Setting UseInversion halves the search space. By default, Alg is Shanks and UseInversion is true. 

(<JacHypPt> P, <RngIntElt> l, <RngIntElt> u, <RngIntElt> n0, <RngIntElt> m) -> RngIntElt
[
Alg: "PollardRho" | "Shanks", 
UseInversion: BoolElt
]

Computes the order of P using either the generic Shanks or a Pollard-Rho variant. The search space must be bounded by l and u and n0 and m are such that |g| (or the group order) is n0 mod m. Setting UseInversion halves the search space. By default, Alg is Shanks and UseInversion is true. 

(<RngQuadIdlOld> I) -> <unknown>

The quadratic order that I is contained in

(<FldOrd> F) -> RngOrd

The order which F is the field of fractions to

(<DB[qm]> D, <RngIntElt> i) -> RngIntElt

The order of the i-th entry of database D

(<DB[qm]> D, <RngIntElt> d, <RngIntElt> i) -> RngIntElt

The order of the i-th entry of dimension d in database D

(<CrvEll> E, <RngIntElt> r) -> RngIntElt

The order of the group of rational points on the degree r extension of the base ring, where E is defined over a finite field.

(<SetPtEll> E, <RngIntElt> r) -> RngIntElt

The order of the group of rational points on the degree r extension of the ring of E, where E is defined over a finite field.

(<CrvEll> E, <FldFin> K) -> RngIntElt

The order the group of rational points on E over K, where E is defined over the rationals or a finite field.

(<JacHypPt> pt) -> RngIntElt

The order of the point on the Jacobian over a finite field or the rationals. Result is zero if point has infinite order.

(<JacHyp> J) -> RngIntElt
[
NaiveAlg,
ShanksLimit,
CartierManinLimit,
UseSchoof,
UseHalving,
UseSubexpAlg
]

The order of the Jacobian J defined over a finite field. 

(<GrpDrch> G) -> RngIntElt

The order of G.

(<GrpDrchElt> x) -> RngIntElt

The order of x.


> g;
q + a*q^2 - 5*q^4 - a*q^5 + 5*q^7 + O(q^8)
> O := Order([Coefficient(g,2), Coefficient(g,3)]);
> O;l
Equation Order with defining polynomial x^2 + 9 over Z
> O;

>> O;
   ^
User error: bad syntax
> O;
Equation Order with defining polynomial x^2 + 9 over Z
> IsGamma0(M);
false
> g;

In file "/home/was/magma/packages/modform/code/q-expansions.m", line 146, column 29:
>>       if IsGamma0(Parent(f));
                               ^
User error: bad syntax

In file "/home/was/magma/packages/modform/code/input_output.m", line 54, column 28:
>>    printf "%o", PowerSeries(f);
                              ^
Runtime error in 'PowerSeries': Intrinsic no longer defined
> IsGamma0(Parent(g));
false
> ;
> g;

In file "/home/was/magma/packages/modform/code/q-expansions.m", line 147, column 32:
>>          bnd := Ceiling(Weight(M) * idxG0(Level(M)) / 12);
                                  ^
Runtime error: Undefined reference 'M' in package "/home/was/magma/packages/modform/code/q-expansions.m"
> IsGamma0(Parent(g));
false
> g;

In file "/home/was/magma/packages/modform/code/q-expansions.m", line 147, column 37:
>>          bnd := Ceiling(Weight(f) * idxG0(Level(f)) / 12);
                                       ^
Runtime error: Undefined reference 'idxG0' in package "/home/was/magma/packages/modform/code/q-expansions.m"
> ;
> IsGamma0(Parent(g));
false
> g;

In file "/home/was/magma/packages/modform/code/q-expansions.m", line 154, column 37:
>>          bnd := Ceiling(Weight(f) * idxG1(Level(f)) / 12);
                                       ^
Runtime error: Undefined reference 'idxG1' in package "/home/was/magma/packages/modform/code/q-expansions.m"
> ;
> g;
q + a*q^2 - 5*q^4 - a*q^5 + 5*q^7 + O(q^8)
> CoefficientRing(g);
Equation Order with defining polynomial x^2 + 9 over Z
> O := $1;
> Discriminant(O);
-36
> G<eps> := DirichletGroup(13,CyclotomicField(6));
> Order(eps);
6
> S := CuspidalSubspace(ModularForms(eps,2));
> S;
Space of modular forms on Gamma_1(13) with character all conjugates of [eps], weight 2, and dimension 2 over Integer Ring.
> S := CuspidalSubspace(ModularForms(eps^3,2));
> S;
Space of modular forms on Gamma_1(13) with character all conjugates of [eps^3], weight 2, and dimension 0 over Integer Ring.
> S := CuspidalSubspace(ModularForms(eps,2));
> S;
Space of modular forms on Gamma_1(13) with character all conjugates of [eps], weight 2, and dimension 2 over Integer Ring.
> Newforms(S);
[* [*
q + (-a - 1)*q^2 + (2*a - 2)*q^3 + a*q^4 + (-2*a + 1)*q^5 + (-2*a + 4)*q^6 + O(q^8),
q + (-b - 1)*q^2 + (2*b - 2)*q^3 + b*q^4 + (-2*b + 1)*q^5 + (-2*b + 4)*q^6 + O(q^8)
*] *]
> CoefficientRing($1[1][1]);
Equation Order with defining polynomial x^2 - x + 1 over Z
> Discriminant($1);
-3
>